Simulation Bias and Variance in Integral Estimates

Why Simulation Error Has More Than One Form

A simulated integral is random because it depends on random draws, even when the quantity being integrated is fixed. For an ordinary average of simulated integrand values, the simulation error has a particularly direct description: its expectation equals the integral, and its variance decreases as the number of independent draws increases. But econometric calculations often use the integral inside another function. Taking its logarithm, for example, can introduce bias even when the integral estimate itself is unbiased. Keeping these two sources of error distinct matters when interpreting simulated likelihood contributions and other transformed quantities.

Here, the integral is treated as fixed, conditional on the data and parameter values at which it is evaluated. The randomness discussed is only the randomness of the simulation draws. It is therefore separate from sampling uncertainty in the observed data or uncertainty about an estimated parameter.

Bias and Variance of a Simulated Integral

Suppose an integral can be written as an expectation \(I=E[h(U)]\), where \(U\) has a specified distribution and \(h(U)\) is the integrand evaluated at a draw from that distribution. Draw \(U_1,\ldots,U_R\) independently from the same distribution, and let \(Y_r=h(U_r)\). The basic simulated estimate is the average of these \(R\) integrand draws.

The standard results require that the draws are independent and identically distributed from the distribution defining the expectation, and that the integrand has finite mean and variance: \(E[h(U)]=I\) and \(\operatorname{Var}(h(U))=\sigma_h^2<\infty\). No assumption that the integrand is normally distributed is needed for the expectation and variance calculations.

Why the Basic Estimate Is Unbiased

The result follows from linearity of expectation and independence across draws. The variance calculation uses independence to eliminate the covariance terms. Showing both steps makes clear which assumption is doing the work.

For any estimator with finite second moment, mean squared error is variance plus squared bias. In this case the bias is zero, so the mean squared error equals the variance. Increasing \(R\) by a factor of four halves the simulation standard error and reduces variance and mean squared error by a factor of four. It does not make any one finite-run estimate equal to the target; rather, it reduces its dispersion across repeated simulations.

When \(\sigma_h^2\) is unknown, it can be estimated from the simulated values \(Y_1,\ldots,Y_R\). Their sample variance is \(s_h^2=(R-1)^{-1}\sum_{r=1}^R(Y_r-\hat I_R)^2\), giving the estimated Monte Carlo standard error \(s_h/\sqrt R\). This estimates the variability due to simulation; it does not measure uncertainty from the original data.

Taking a Logarithm Introduces a Different Bias

An unbiased estimate does not remain unbiased after an arbitrary nonlinear transformation. Let \(I>0\), and suppose \(Y_r=h(U_r)>0\) almost surely, so that \(\hat I_R>0\) and its logarithm is defined. If the logarithm is integrable, Jensen's inequality for the concave function \(\log(x)\) gives

Therefore, the plug-in estimate \(\log(\hat I_R)\) is generally downward biased for \(\log(I)\). The inequality is strict when \(\hat I_R\) is nonconstant and the relevant expectations exist. The source of this bias is the curvature of the logarithm, not bias in the simulated integral estimate. In contrast, \(\log(I)\) is the log of the target integral; it is not the average of the logarithms of the simulated integrand values.

If the integrand can be zero, a finite-run estimate may also be zero, in which case its logarithm is undefined. A log-transformation analysis therefore needs to address positivity as well as bias. Positivity of every simulated value ensures a positive finite-run average, though conditions ensuring that the expected log exists are still needed for an expectation-based bias statement.

Approximating the Bias and Variance of the Log Estimate

For a larger number of draws, a Taylor expansion gives useful approximations. Write \(\delta=\hat I_R-I\), so that \(E[\delta]=0\) and \(E[\delta^2]=\sigma_h^2/R\). When \(I>0\), expand the logarithm around \(I\):

Taking expectations removes the first-order term because \(E[\delta]=0\). Under regularity conditions that make the expected higher-order remainder negligible relative to \(1/R\), such as bounded integrand values bounded away from zero, the leading bias is

This approximation identifies the scale of the bias: it is approximately proportional to the variance of the integral estimate and inversely proportional to the square of the integral. It is a large-\(R\) approximation, not an exact correction for every finite number of draws.

For the leading variance approximation, retain the first-order term \(\delta/I\). Its variance is \(\operatorname{Var}(\delta/I)=\sigma_h^2/(RI^2)\). Under the same type of regularity conditions, this yields

The corresponding approximate standard error is \(\sigma_h/(\sqrt R I)\). In practice, replacing \(\sigma_h\) with the sample standard deviation \(s_h\) gives an estimated simulation standard error. These approximations describe how the random log estimate varies around its own expectation; they do not remove its downward bias.

What These Results Do and Do Not Say

For the basic simulated average, the finite-sample result is exact: under independent draws with the correct distribution and finite integrand variance, it is unbiased and its variance is \(\sigma_h^2/R\). The log-bias result is different. Jensen's inequality establishes the direction of bias under its stated conditions, while the Taylor calculation supplies a leading approximation under additional regularity conditions.

These distinctions are useful whenever an integral is an intermediate quantity. One can report the simulated integral estimate and its Monte Carlo standard error, then separately consider the effect of using a transformation such as a logarithm. Increasing the number of draws reduces the variance of the integral estimate and, in the regular setting above, reduces the leading log bias at rate \(1/R\). It does not turn the log of a finite-run estimate into an exactly unbiased estimate of the log integral.